Question

If a \({\bf{3}} \times {\bf{8}}\) matrix A has a rank 3, find dim Nul A, dim Row A, and rank \({A^T}\).

Short answer

5, 3, and 3

Step-by-step solution

Find dim Nul A

Using the rank theorem,you get:

\(\begin{aligned} {\rm{rank}}\,A + \dim \,{\rm{Nul}}A &= n\\3 + \dim \;{\rm{Nul}}\,A &= 8\\\dim \;{\rm{Nul}}\,A &= 8 - 3\\ &= 5\end{aligned}\)

Find dim row A

The dim row A is equal to the rank of A i.e., 3.

Find the rank of \({A^T}\)

\(\dim \,\;{\rm{Row}}\;A = \dim \,{\rm{Col}}\,{A^T} = 3\)

The rank of \({A^T}\) is equal to dim Col \({A^T}\); so the rank of \({A^T}\) is 3.

Thus, dim Nul A =5, dim row A=3, and the rank of \({A^T}\)=3.